Properties

Label 4016.61
Modulus $4016$
Conductor $4016$
Order $500$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4016, base_ring=CyclotomicField(500))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,375,306]))
 
pari: [g,chi] = znchar(Mod(61,4016))
 

Basic properties

Modulus: \(4016\)
Conductor: \(4016\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(500\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4016.bs

\(\chi_{4016}(29,\cdot)\) \(\chi_{4016}(37,\cdot)\) \(\chi_{4016}(53,\cdot)\) \(\chi_{4016}(61,\cdot)\) \(\chi_{4016}(77,\cdot)\) \(\chi_{4016}(109,\cdot)\) \(\chi_{4016}(133,\cdot)\) \(\chi_{4016}(141,\cdot)\) \(\chi_{4016}(165,\cdot)\) \(\chi_{4016}(213,\cdot)\) \(\chi_{4016}(229,\cdot)\) \(\chi_{4016}(269,\cdot)\) \(\chi_{4016}(277,\cdot)\) \(\chi_{4016}(285,\cdot)\) \(\chi_{4016}(293,\cdot)\) \(\chi_{4016}(333,\cdot)\) \(\chi_{4016}(341,\cdot)\) \(\chi_{4016}(349,\cdot)\) \(\chi_{4016}(381,\cdot)\) \(\chi_{4016}(397,\cdot)\) \(\chi_{4016}(413,\cdot)\) \(\chi_{4016}(421,\cdot)\) \(\chi_{4016}(429,\cdot)\) \(\chi_{4016}(437,\cdot)\) \(\chi_{4016}(453,\cdot)\) \(\chi_{4016}(461,\cdot)\) \(\chi_{4016}(485,\cdot)\) \(\chi_{4016}(493,\cdot)\) \(\chi_{4016}(557,\cdot)\) \(\chi_{4016}(573,\cdot)\) ...

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: $\Q(\zeta_{500})$
Fixed field: Number field defined by a degree 500 polynomial (not computed)

Values on generators

\((2511,3013,257)\) → \((1,-i,e\left(\frac{153}{250}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 4016 }(61, a) \) \(-1\)\(1\)\(e\left(\frac{21}{500}\right)\)\(e\left(\frac{31}{100}\right)\)\(e\left(\frac{69}{250}\right)\)\(e\left(\frac{21}{250}\right)\)\(e\left(\frac{441}{500}\right)\)\(e\left(\frac{317}{500}\right)\)\(e\left(\frac{44}{125}\right)\)\(e\left(\frac{36}{125}\right)\)\(e\left(\frac{3}{500}\right)\)\(e\left(\frac{159}{500}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4016 }(61,a) \;\) at \(\;a = \) e.g. 2